Triangle is a geometrical plane figure. The shape of triangle looks like the figure below:

A triangle has three sides and three angles. It is a closed figure with three line segments as its boundary. These line segments are called sides. It has three corners called vertices. The symbol for a triangle is Δ

The three vertices are generally named by three consecutive alphabets such as A, B and C or X, Y and Z. A triangle with vertices A, B, and C is denoted by ΔABC.

Tri means three and so a Triangle has three angles. These angles are formed by two sides at each of the corners. The angle formed at the vertex A by the two sides AB and AC is called

The three sides and three angles of a triangle together are called the six parts or elements of the triangle.

Now look at the diagram below

The side BC is opposite to vertex A and is also denoted in short by “a”. The sides opposite to B and C are written as b and c. Thus sides BC= a, AC= b and AB= c

**Properties of a triangle**

- The sum of the three angles of any triangle is always 180 degree. <A+ <B + <C =180°
- Any angle of a triangle is always greater than zero and less than 180°
- Not more than one angle can be 90° or more.
- The sum of length of any two sides will always be greater than the length of the third side.
- It is a closed plain figure.

## Types of Triangles

A triangle can be classified by either its sides: or its angle.

**(A) A triangle is classified in three types as scalene, isosceles, or equilateral triangle based on its sides.**

**1. Equilateral triangle**

A triangle having all sides equal is called an equilateral triangle. It’s all angles are also equal. As sum of three angles of a triangle is 180°, so each angle of an equilateral triangle is 60°.

**2. Isosceles triangle **

A triangle having two sides equal is called an isosceles triangle. The angles opposite to equal sides are also equal.

**3. Scalene triangle **

A triangle having all sides of different length is called scalene triangle.

In the figure above, the sides a≠ b≠ c . Hence this is a scalene triangle.

**(B) A triangle is classified in three types as acute, right, or obtuse triangle based on its angles:**

**1. Acute triangle**

A triangle having all angles less than 90° is called acute triangle.

In the above triangle, the three angles are 50°, 60° and 70°. All three are less than 90°. Hence it is an acute triangle.

**2. Obtuse triangle**

A triangle having one angle more than 90° is called an obtuse triangle.

In the triangle above, the three angles are 25°, 35° and 120°. One of angle is more than 90°. Hence it is an obtuse triangle.

**3. Right triangle**

If anyone angle of a triangle is 90° , the triangle is called right triangle. The triangle below is a right angle triangle as <ACB is right angle (90°). As sum of three angles of a triangle is 180°, in a right triangle sum of the other two angles will be 90°.

The side opposite to the right angle is called the **hypotenuse.** The hypotenuse is larger than the other two sides. In diagram above, the AB is the hypotenuse and is opposite to the right angle <ACB.

The right triangle has a special property- known as the Pythagorean theorem. The square of length of hypotenuse is equal to the sum of the squares of the lengths of the other two sides. . The Pythagorean theorem is named after the Greek mathematician Pythagoras, who proved this relationship during 570-495 BC. The theorem has a wide range of uses in higher mathematics.

**Points to remember about triangles**

- A triangle has three sides and three angles
- A triangle is a plane figure closed by three line segments.
- A triangle has three corners called vertices.
- The sum of the three angles of a triangle is always 180 degree.
- The sum of length of any two sides is always greater than the length of the third side.
- A triangle can be classified by either its sides: or its angle.
- A triangle is classified as scalene, isosceles, or equilateral triangle based on its sides.
- A triangle is classified as acute, right, or obtuse triangle based on measurement of its angles
- A triangle with all sides equal is called an equilateral triangle.
- A triangle with two sides equal is called an isosceles triangle.
- A triangle with all sides of different length is called scalene triangle.
- A triangle with a right angle (90°) is called right triangle.
- A triangle with all angles less than 90° is called acute triangle.
- A triangle having one angle more than 90° is called an obtuse triangle.
- Pythagorean Theorem states that the square of length of hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

## Questions and Answers

**Question 1:**

- Name the above triangle
- How many vertices (corners) the triangle has? Name them
- How many sides the triangle has? Name them
- How many angles the triangle has? Name them

**Solution:**

- The triangle is ΔXYZ
- The triangle has three vertices X, Y and Z
- The triangle has three sides XY, XZ and YZ
- The triangle has three angles <X, <Y and <Z

**Question 2 : **What are the elements (parts) of ΔPQR?

**Solution: **ΔPQR has three sides and three angles. The three sides are PQ, QR and PR and the three angles are <P, <Q and < R.

**Question 3: **The two angles of a triangle are 60° and 45°. What is the measurement of third angle?

**Solution: **Let** **<A = 60° and <B =45° . Then <C = ?

The sum of the three angles of any triangle is always 180 degree.

<A+ <B + <C =180°

<C =180° – <A- <B = 180-60-45 = 75°

**Question 4 : **Below are three triangles. Classify them into an acute triangle, obtuse angle and right angle?

**Solution:**

- One angle of Triangle 1 is a right angle (90°) . So Triangle 1 is a right triangle.
- All angles of Triangle 2 are less than 90°. So Triangle 2 is an acute angle.
- One angle in Triangle 3 is more than 90°. So the Triangle 3 is an obtuse triangle.

**Question 5:** What is the Pythagorean Theorem?

**Solution:**

The Pythagorean Theorem states that the square of length of hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides. . This can be written also as

## Exercise

1. How many parts a triangle has?

- 3
- 6
- 7

2. If two angles of a triangle are 30° and 45°, what is measurement of third angle?

- 60°
- 90°
- 105°

3. If each angle of a triangle is equal to 60°, identify the triangle.

- Right triangle
- Equilateral triangle
- Isosceles triangle

4 . If an angle of a triangle is equal to 120°, identify the triangle.

- Obtuse triangle
- Acute triangle
- Right triangle

#### True or false

6. If an angle of a right triangle is 45°, it will also be an equilateral triangle

- True
- False

7. A triangle can have more than one right angle

- True
- False

7. An isosceles triangle can be right triangle also

- True
- False

#### Fill in the blank

8. A/An ………………….. triangle has two sides equal

- Right triangle
- Obtuse triangle
- Isosceles triangle

9. A triangle can not have more than ……….. obtuse angle

- 0
- 1
- 2

10. Which of the following statements is NOT true for a right triangle?

- The length of hypotenuse is equal to the sum of the lengths of the other two sides
- The length of hypotenuse is equal to the sum of the squares of the lengths of the other two sides
- The square of length of hypotenuse is equal to the sum of the squares of the lengths of the other two sides

Muhammad fakhir yasir says

Very good exercise .

I am very happy

Mayra khan says

Nice

stephano says

Thanks alot for this

Thomas Johnston says

Very worthwhile review!